Let $f(x)=x^{^{\scriptsize\dfrac{1}{2}}}$. $f'(4)=$
Explanation: Let's first find the expression for $f'(x)$ and then evaluate it at $x=4$. The derivative of $f$ can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a fraction.) $\begin{aligned} &\phantom{=}f'(x) \\\\ &=\dfrac{d}{dx}\left(x^{^{\frac{1}{2}}}\right) \\\\ &=\dfrac{1}{2}x^{^{\frac{1}{2}-1}} \gray{\text{The power rule}} \\\\ &=\dfrac{1}{2}x^{^{-\frac{1}{2}}} \end{aligned}$ So we found that $f'(x)=\dfrac{1}{2}x^{^{-\frac{1}{2}}}$, which can also be written as $\dfrac{1}{2\sqrt{x}}$ Now let's plug ${x=4}$ : $\begin{aligned} \dfrac{1}{2\sqrt{ 4}}&=\dfrac{1}{2\cdot 2} \\\\ &=\dfrac14 \end{aligned}$ In conclusion, $f'(4)=\dfrac{1}{4}$.